Multiply.
$\frac{x^2-4x+3}{x^2+3x+2} \cdot \frac{3x+3}{x-1}$
Simplify your answer as much as possible.
To simplify the expression \(\frac{x^2 - 4x + 3}{x^2 + 3x + 2} \cdot \frac{3x + 3}{x - 1}\), we begin by factoring each part of the rational expressions.
The original expression is then rewritten as:
\[\frac{(x-1)(x-3)}{(x+1)(x+2)} \cdot \frac{3(x+1)}{x-1}\]
Now, we combine the fractions into a single product:
\[\frac{(x-1)(x-3) \cdot 3(x+1)}{(x+1)(x+2) \cdot (x-1)}\]
By identifying the common factors in the numerator and the denominator, we can simplify the expression. The factor \((x-1)\) appears in both the numerator and the denominator, and the factor \((x+1)\) also appears in both. Assuming \(x
eq 1, x
eq -1, x
eq -2\), these factors cancel out:
\[\frac{\cancel{(x-1)}(x-3) \cdot 3 \cancel{(x+1)}}{\cancel{(x+1)}(x+2) \cdot \cancel{(x-1)}} = \frac{3(x-3)}{x+2}\]
Thus, the simplified expression is \(\frac{3x-9}{x+2}\).
Simplifying rational expressions is an essential skill in algebra that allows us to manage complex relationships between variables. Imagine you are designing a budget for a project where the cost is determined by a ratio of different components, such as \(\frac{x^2-4x+3}{x^2+3x+2}\) multiplied by another factor \(\frac{3x+3}{x-1}\). By simplifying the formula, you can calculate the total cost much faster and minimize the risk of errors in your planning. This mathematical technique helps in various fields, from engineering calculations to optimizing resource distribution, by turning complicated expressions into clear and manageable ones.